Extensions of an Abelian-Higgs model have been largely considered as a toy model in order to build theories beyond the Standard Model (SM). As it is well-known, such a model supports classical vortices [1]} with very precisely known solutions [2]}. It seems then relevant to analyze whether such configurations still hold and under what circumstances once the Abelian-Higgs model is modified by the inclusion of portal terms. Moreover
vortices could play an important role as dark strings. Indeed, cosmic string networks are are hypothetical one dimensional topological defects that could be formed during phase transitions in the early universe [3]}, [4]}, [5]}, [6]}, [7]}. Then, string radiation might lead to a sizable population of non-thermal dark matter [8]}, [9]}, [10]}. This mechanism has been particularly studied in connection with the existence of axions [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, WIMPs [17]}, [18]} and hidden photons [19]}.
